Density in Metric Spaces
An alternative definition of dense set in the case of metric spaces is the following. When the topology of X is given by a metric, the closure of A in X is the union of A and the set of all limits of sequences of elements in A (its limit points),
Then A is dense in X if
Note that . If is a sequence of dense open sets in a complete metric space, X, then is also dense in X. This fact is one of the equivalent forms of the Baire category theorem.
Read more about this topic: Dense Set
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“When I consider the short duration of my life, swallowed up in the eternity before and after, the little space which I fill and even can see, engulfed in the infinite immensity of spaces of which I am ignorant and which know me not, I am frightened and am astonished at being here rather than there. For there is no reason why here rather than there, why now rather than then.”
—Blaise Pascal (1623–1662)